`WarpKriging::logLikelihood`

Description

Get the Maximised Log-Likelihood of a WarpKriging Model Object.

Usage

Python

# wk = WarpKriging(...)
wk.logLikelihood()

R

# wk <- WarpKriging(...)
wk$logLikelihood()

Matlab/Octave

% wk = WarpKriging(...)
wk.logLikelihood()

Julia

# wk = WarpKriging(...)
ll = logLikelihood(wk)

Arguments

Argument	Description
None	`logLikelihood()` reads the value attached to the fitted model.

Details

Returns the concentrated profile log-likelihood at the fitted warp parameters and \(\hat\theta\). The variance \(\hat\sigma^2\) and trend coefficients \(\hat\beta\) are computed analytically from \(R(\theta)\) and \(y\):

\[ \hat\beta = (F^\top R^{-1} F)^{-1} F^\top R^{-1} y, \quad \hat\sigma^2 = \tfrac{1}{n}(y - F\hat\beta)^\top R^{-1} (y - F\hat\beta), \]

and the concentrated log-likelihood is

\[ \ell_{\text{prof}}(\theta, w) = -\tfrac{n}{2}\bigl[1 + \log(2\pi) + \log\hat\sigma^2\bigr] - \tfrac12 \log |R|. \]

Value

A numeric scalar: the maximised log-likelihood of the fitted WarpKriging model.

Examples

f <- function(x) 1 - 1 / 2 * (sin(12 * x) / (1 + x) + 2 * cos(7 * x) * x^5 + 0.7)
X <- as.matrix(seq(0.05, 0.95, length.out = 10))
y <- f(X)

wk <- WarpKriging(
  y, X,
  warping = "kumaraswamy",
  kernel = "gauss",
  parameters = list(max_iter_adam = "20", max_iter_bfgs = "10")
)
print(wk)
print(wk$logLikelihood())

Results

* WarpKriging
* data: 10x[0.05,0.95] -> 10x[0.163421,0.976851]
* trend constant (est.): 126.685
* variance (est.): 2.63805e+08
* covariance:
  * kernel: gauss
  * range (est.): 9
  * warpings:
      x0: "kumaraswamy"  →  Kumaraswamy(a=1.01912, b=0.981236)
  * total warp params: 2
  * fit:
    * objective: LL
    * optim: BFGS+Adam
[1] -24.37993

WarpKriging::logLikelihood